Java 2 Platform, Standard Edition (J2SE) provides a complete environment for applications development on desktops and servers. It also serves as the foundation for the Java 2 Platform, Enterprise Edition (J2EE) and Java Web Services.
Java 2 Platform, Enterprise Edition (J2EE) defines the standard for developing component-based multitier enterprise applications. Features include Web services support and development tools (SDK).
For more details, you should visit Sun's J2SE and J2EE pages.
Of course, I'm sure someone here will also add something from their own point of view. However, if you can't wait, follow the above links.
In simplistic terms, J2SE (Java 2 Standard Edition) is the Java programming language and its core API's (Application Program(ming) Interface). J2EE (Java 2 Enterprise Edition) is a group of API's that can be used in addition to (or "on top of") J2SE. It contains API's that are usually used for larger applications, multi-tiered (or layered) applictions or server side applications. It contains API's (such as the Java Database Connect (JDBC), Servlet, JSP, Enterprise JavaBeans, and Java Mail API's as some examples) that are not part of J2SE and are therefore not contained in the Java Runtime Environment (JRE)installed on a typical client. Many J2EE API's are implemented by third parties in things such as Web Servers, Application Servers, Database Connection Libraries, etc.